Data processing with magnetic resonance tool

ABSTRACT

Various embodiments include apparatus and methods to acquire echo signals associated with operation of a nuclear magnetic resonance logging tool to evaluate a formation and process the echo signals taking into account motion of the nuclear magnetic resonance logging tool. Coefficients may 5 be generated that are correlated to porosity of the formation. Additional apparatus, systems, and methods are disclosed.

CLAIM OF PRIORITY

This application claims the benefit of U.S. Provisional Application Ser.No. 61/806,279, filed on Mar. 28, 2013 which application is incorporatedby reference herein in its entirety.

TECHNICAL FIELD

The present invention relates generally to apparatus and methods relatedto nuclear magnetic resonance.

BACKGROUND

Nuclear magnetic resonance (NMR) is used as a tool in a number ofdifferent technology areas to investigate different types of mediums.NMR can occur when the medium is subjected to a static magnetic field,B₀, and to an oscillating magnetic field, B₁. When subjected to anapplied static magnetic field, polarization of nuclear magnetic spins ofthe medium occurs based on spin number of the medium and magnetic fieldstrength. Applying an electromagnetic field to the medium in the staticmagnetic field can perturb the polarization established by the staticmagnetic field. In optimal measurements, the static magnetic field andthe perturbing field are perpendicular to each other. Collectedresponses received from the medium related to the total magnetization ofnuclear spins in the medium, in response to these applied fields, can beused to investigate properties of the medium, and may provide imaging ofthe medium. It is noted that magnetization is proportional topolarization.

Nuclear magnetic resonance measurements are created by the oscillationof excited nuclear magnetic spins in the transverse plane, that is, thedirection perpendicular to the magnetic field. This oscillationeventually dies out and the equilibrium magnetization returns. Thereturn process is referred to as longitudinal relaxation. The timeconstant, T₁, for nuclei to return to their equilibrium magnetization,M_(o), is called the longitudinal relaxation time or the spin latticerelaxation time. The magnetization dephasing, that is losing coherence,along the transverse plane is given by the time constant T₂ and iscalled the spin-spin relaxation time. The loss of phase coherence can becaused by several factors including interactions between spins,electrons, or magnetic gradients.

A widely used NMR measurement technique, designed by Carr, Purcell,Meiboom, and Gill and, hence, referred to as CPMG, uses a sequence ofradio frequency pulses to produce spin echoes and counteract dephasingof the magnetization in the medium investigated. In the CPMG sequence,an initial pulse, commonly a 90° pulse, can be applied to tip thepolarization into a plane perpendicular to the static magnetic field. Tocounter dephasing due to magnetic inhomogeneities, another pulse, arecovery pulse, commonly a 180° or other angle tipping pulse, is appliedto return to phase, which produces a signal called an echo from themedium. Yet, after each return to phase, dephasing begins and anotherrecovery pulse is applied for rephasing. Rephasing or refocusing isrepeated many times in the CPMG sequence, followed by measuring eachecho.

The echo magnitude decreases with time due to a number of irreversiblerelaxation mechanisms. The CPMG sequence can have any number of echoes,where the time between each echo can be relatively short, for example,of the order of 0.5 ms or less or as long as 12 ms is used.

NMR logging tools have long proven their value to formation evaluation.Petrophysical information can be derived from NMR measurements, such as,but not limited to petrophysical properties of fluid containing porousmedia. Various properties that can be measured using an NMR logging toolinclude pore size, porosity, surface-to-volume ratio, formationpermeability, and capillary pressure. These properties are determinedfrom inversion of data. Recently, new drilling tools have addedlow-gradient magnet configurations to help reduce the effects of axismotion. The primary challenge associated with using low-gradient toolsis the requirement of one preferred sensitive volume to be tracked overtemperature. The secondary challenge is that the sensitive volumeassociated with low-gradient tools provides a vertically short sensitivevolume. As a result, the tools are more sensitive to vertical motion,and thus to rate of penetration (ROP) or pulling speed, opposed tohigh-gradient configuration tools, particularly for T₁ logging. Not onlyis porosity affected, but the T₁ spectrum can also be distorted. Havinga more reliable inversion may provide more precision in the evaluationof NMR data to generate correct porosity, T₂ spectra, T₁ spectra,diffusion spectra, and other parameters.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a representation of a sample with non-zero nuclear magneticspin of a nuclear magnetic resonance experiment.

FIG. 2A is a representation of magnetic spins aligning with a magneticfield according to the Boltzmann distribution when placed in themagnetic field.

FIG. 2B is a representation of a second time varying magnetic fieldapplied as an excitation.

FIG. 2C is a representation of a signal received from the excitednuclear spins according to the principle of reciprocity.

FIG. 3 is a representation of the interactions of a static magneticfield and a radio frequency field magnetic field through nuclear spin ina nuclear magnetic resonance experiment.

FIG. 4 is a representation of a system in equilibrium with themagnetization aligned with a static magnetic field and rotating on itsaxis around the direction of the static magnetic field.

FIG. 5 is a representation of two characteristics of magnetization:longitudinal recovery and transverse relaxation.

FIG. 6 is a representation of a sequence known as the Carr, Purcell,Meiboom, and Gill sequence.

FIG. 7 is a diagram of a T₁ experiment.

FIG. 8 is a demonstration of a nuclear magnetic resonance tool beingpulled out of a borehole, while conducting nuclear magnetic resonancemeasurements.

FIG. 9 is a representation of a Carr, Purcell, Meiboom, and Gillsequence following the T₁ recovery of the nuclear magnetic resonancetool of FIG. 8 being pulled out of a borehole, while conducting nuclearmagnetic resonance measurements.

FIG. 10 is a representation of A₀ being artificially high for mediumwait times caused by over polarization.

FIG. 11 is a representation of a result from a pseudo integration of allthe magnetic field felt until the time of interest in conducting nuclearmagnetic resonance measurements.

FIG. 12 is a representation of additional phase scrambling in nuclearmagnetic resonance measurements due to pulling the tool in the borehole.

FIG. 13 is a representation of a field of a tool along its axis as aone-dimensional line.

FIG. 14 is a representation of a decrease in echo amplitude as the echotrain proceeds due to imperfect radio frequency field magnetic field.

FIG. 15 is a pictorial representation of a grid mesh field.

FIG. 16 is an example a simulated echo train.

FIG. 17 is a representation of where T₁ points should be in a plot ofsignal versus experiment T₁ time for an example fast ROP experiment.

FIG. 18 is an example of a plot of porosity versus T₁ for a high ROPwhen standard the inversion is used.

FIG. 19 is an example of a plot of intensity versus T₁ of theoreticaldata using the new inversion method.

FIG. 20 is a representation showing the porosity vs T1 when using thenew inversion method.

FIG. 21 is a representation of an excited volume, which happens at thetime of the excitation pulse.

FIG. 22A is a representation of a T1 sequence showing the movement ofthe sensitive region relative to a nuclear magnetic resonance tool,where the sensitive volume at the excitation pulse is no longer thesensitive volume that had the saturation pulse applied.

FIG. 22B is a representation of a T₁ experiment considering only a freshzone and no over-polarization effects.

FIG. 23A is a representation of a magnetic field at constant radialdistance away from a tool and the magnetization that field creates whenmoving at a constant speed.

FIG. 23B is a representation of the effects of the magnetization having“memory” known as over polarization.

FIG. 24A is a pictorial of a selected sensitive region in a magneticfield.

FIG. 24B is a representation of a rotationally symmetric voxel.

FIG. 25 is a flow diagram of features of an example method to processnuclear magnetic resonance data taking into account motion of thenuclear magnetic resonance logging tool.

FIG. 26 is a flow diagram of features of an example method to processnuclear magnetic resonance data taking into account motion of a nuclearmagnetic resonance logging tool.

FIG. 27 is a flow diagram of features of an example method to processnuclear magnetic resonance data taking into account motion of a nuclearmagnetic resonance logging tool.

FIG. 28 is a block diagram of features of an example system operable toprocess nuclear magnetic resonance data taking into account motion of anuclear magnetic resonance logging tool, in accordance with variousembodiments.

DETAILED DESCRIPTION

The following detailed description refers to the accompanying drawingsthat show, by way of illustration and not limitation, variousembodiments in which the invention may be practiced. These embodimentsare described in sufficient detail to enable those skilled in the art topractice these and other embodiments. Other embodiments may be utilized,and structural, logical, and electrical changes may be made to theseembodiments. The various embodiments are not necessarily mutuallyexclusive, as some embodiments can be combined with one or more otherembodiments to form new embodiments. The following detailed descriptionis, therefore, not to be taken in a limiting sense.

In various embodiments, the accuracy and precision of NMR inversion fordownhole logging data is increased. One or more processors and memorysystems can be used to execute instructions to realize accuracy andprecision of NMR inversion for downhole logging data. Data can beprovided by NMR tools.

A NMR experiment can be conducted on a sample with non-zero nuclearmagnetic spin represented in FIG. 1. When placed in a magnetic field themagnetic spins tend to align more towards that magnetic field than not,according to the Boltzmann distribution as shown in FIG. 2A. Thismagnetic field traditionally is referred to as the “main magnetic field”or the “static field” as it is usually independent of time and is giventhe symbol B₀. The bulk effect of nuclear magnetic spin alignment iscalled the magnetization and given the symbol M. It can be thought of asmini localized magnetic fields.

A second time varying magnetic field is also applied as shown in FIG.2B. That field is designated as B₁ (also called the “radio frequencyfield”) and is turned on and off at different increments, known as apulse. This magnetization excitation results in a tip angle, θ_(Tip).The B₁ magnetic field in the majority of NMR experiments is created by acoil, called an antenna, and is powered by a “transmitter.”

The NMR experiment depends on the nuclear spins (the object of interest)interaction with the B₁ and B₀ as shown in FIG. 3. The B₀ direction atany point dominates the interaction when B₀>>B₁, which is the case fordownhole tools. This means only portions perpendicular to the B₀ willmatter for B1. The last part of the NMR experiment is to receive asignal from the excited nuclear spins. This happens according to theprinciple of reciprocity as shown in FIG. 2C, as the nuclear spins actas mini transmitting coils. The system in equilibrium will looksomething like that of FIG. 4. The local magnetization is aligned withthe local B₀ magnetic field and rotates on its axis around the B₀'sdirection.

The received data, in contemporary downhole experiments, reveals twocharacteristics of the magnetization: longitudinal recovery (T₁) andtransverse relaxation (T₂) shown in FIG. 5. Since the magnetic fields inlogging tools have inhomogeneities greater than the parts per a million,which is required to reveal Larmor frequencies of different nuclearactive sites, the distinguishing chemical properties that can be foundare only T₁, T₂, and sometimes a diffusion coefficient (D).

Both of these measurements require the magnetization to be perturbedfrom its equilibrium state, alignment with the static field. These aremeasurements of the magnetizations rates to equilibrium from theperturbed states in the direction parallel and perpendicular to the B₀field. A third return to equilibrium is possible known as T_(rho), butis not commonly used downhole at this time.

The magnetization can be manipulated in order to cause the perturbationby the B₁ such that the measurements above for T₁ and T₂ can be made.Two specific manipulations are the main focus of most NMR experimentsdownhole. Other more complicated manipulations (rotation) are possibleand can be useful, but only the two specific manipulations arediscussed, since they are the most common. The first manipulation is amanipulation such that the magnetization is in the perpendicular plane,commonly called a “90” or “π/2” pulse. The second manipulation is an“inversion,” that is to say, to cause the magnetization to point in theopposing direction. From equilibrium, the opposing direction would be inthe anti-parallel direction. This pulse is commonly referred to as a“180” or “π” pulse. By timing the pulse length, or changing itsamplitude, the magnetization can be rotated any amount.

The T₂ of the formation is heavily influenced by the formation andgradient of the tool. If a free induction decay (FID) were measured andthe T₂ calculated would not be the intrinsic T₂, but a modified oneknown as T₂*. A truer T₂ is achieved point by point by refocusing themagnetization with a sequence of 180 pulses, this T2 is still subjugatedto irreversible losses due to diffusion and formation surface effects.The FID is then known as an Echo FID. The Echo FID is integrated to givea single value (called an Echo) and creates an Echo train. FIG. 6 is arepresentation of this sequence, which is known as the Carr, Purcell,Meiboom and Gill sequence (CPMG).

FIG. 6 illustrates use of a saturation pulse, a 90° tipping pulse, and asequence of 180° refocusing pulses. In this non-limiting examplesequence, ten 180° refocusing pulses cause ten echoes 607-1 . . .607-10, where the peak amplitudes of the echoes are equally spaced apartby a peak to peak time distance, TE (echo time), that corresponds to theequally spaced apart time distances of the refocusing pulses. Refocusingpulses are not limited to ten pulses, but the number used may depend onthe application and/or measurement parameters. Also indicated are anacquisition windows 609-1 . . . 609-10 for capturing the signal of anecho, a first echo E₁, a second echo E₂, a third echo E₃, and A₀. A₀ isthe amplitude of the echo train at time zero. A₀ can be calculated byusing an exponential decay fitting curve determined from a third echo E₃to the last echo. E₁ and E₂ can be included if they are corrected. Theseechoes decay according to the T₂ of the medium. Once the nuclear spinpopulation is fully recovered for the sequence, the medium can be probedagain by another sequence.

A T₁ experiment downhole consists of a flipping or nulling of themagnetization in the positive z direction through a 180, 90, orsaturation/inversion pulse followed by a CPGM sequence. The time betweenthe nulling pulse and CPMG, designated as wait time (WT), is varied inthe τ time domain. This allows for the built up magnetization in the zaxis to be measured. Any number of wait times can be used with a minimumin a range of 2 to about 10; for example in practice, a minimum of 5 isused. There is no upper limit on how many WTs can be used; however, itis preferred to keep the number lower so that the vertical resolution ofthe data is kept minimal.

FIG. 7 is a diagram of a T₁ experiment. The CPGM can have any number ofechoes, and vary the number of echoes as the WT varies. Commonly, thelongest wait time can have a significantly larger number of echoestaken. Lower WTs don't need as many echoes as less magnetization hasrecovered and the data decays into the noise quickly.

The NMR echo signal is commonly calculated by the equation:

$\begin{matrix}{{{s(t)} = {\left( {1 - ^{- \frac{WT}{T_{1}}}} \right) \cdot ^{- \frac{t}{T_{2}}} \cdot ^{{- {D{({\gamma \; G\; {TE}})}}^{2}}{t/12}}}},} & (1)\end{matrix}$

where WT (wait time) is the time allowed for the magnetization topolarize, T₁ is the longitudinal recovery time constant, t is the timeof the echo peak, T₂ is the transverse decay constant, D is thediffusion constant, γ is the gyromagnetic ratio, G is the gradient, andTE is the time between echoes.

The additional complexity in the NMR experiment, while logging, is themotion of the tool. The logging tool is pulled through the formation orcould be on the back of a drilling string. In downhole logging, toolsare pulled through the formation at speed typically between 30 ft/hr to120 ft/hr. For wireline, the speed of pulling could be as high as 720ft/hr. The rate at which a tool is pulled is referred to as a rate ofpenetration (ROP). This motion has several potential effects on the NMRecho trains to deviate it from equation (1), which causes the inversionto be incorrect. To the CPMG sequence, there are losses due to formationmoving into a new zone while pulsing, in the T₁ experiment there can beover-call due to fresh zones after the saturation/inversion pulse, thereare pre-polarization motion dependent effects due to magnetic fieldshape, a de-phasing, and additional rotation errors due to imperfect B₁.

Consider motion effects with respect to the CPMG. The CPMG has one“excitation” pulse. Commonly this is referred to as the “90” pulse. A 90degree pulse is desired in order to get the most signal from theformation. As discussed below, this pulse may not be a true 90 pulse atall locations. Excited zone 822 is the only zone that will give signalfrom the recovery pulses. FIG. 8 is a demonstration of a tool 805 beingpulled out of a borehole, while conducting NMR measurements. As the tool805 moves, the 180 pulses are used on a zone 832 that was not excited bythe 90 pulse. This means that signal is progressively lost. In theinversion, this does not cause an error to A0 (maximal signal) but willshift the inversion T₂ spectra such that the T₂ appear smaller.

Consider motion effects with respect to the T₁ Recovery. The T₁experiment starts with a saturation/inversion pulse. This pulse isdesigned to “kill” the magnetization. This could be thought of as ascrambling of the magnetization into all orientations or simply puttingit into the transverse plane. Directly after the saturation/inversionpulse no signal should be achievable. As T₁ experiments are an inquiryas to how fast the magnetization builds in the B₀ direction, severalexperiments are done to vary the allotted recovery time, WT, after thesaturation/inversion pulse. Following the recovery is a CPMG sequence,as shown in FIG. 9. The CPMG sequence will suffer the same losses asdescribed above. During the wait time, the tool moves and so does thevolume of inquiry. When the 90 pulse is applied some of the excited zoneis not touched by the previous saturation/inversion pulse and will be100% polarized. That over polarization causes the A₀ to be artificiallyhigh for medium wait times as shown in FIG. 10. At short wait times thetool has not moved much so the effect is not seen and long wait timesshould be 100% polarized so the motion has no effect.

Considering prepolarization, the majority of logging tools have magnetsthat are stronger preceding the NMR zone. The magnetization will relaxaccording to the field strength it feels.

M(t _(i+1))=M _(i)+(χB ₀(r,z)−M _(i))·(1−e ^(−Δt/T1))  (2)

It builds up according to the field it is currently in, and how muchmagnetization is currently there. Essentially, it is a pseudointegration of all the magnetic field it has felt until the time ofinterest. This means magnetization could be higher or lower than if itwere polarizing in a stationary magnetic field, M=χ□₀(1−e^(−Δt/T1)).This causes the possible amount of signal to be different from thestationary case. See FIG. 11.

The diffusion effect in NMR is generally due to the random motion thatoccurs in the formation. It causes an irreversible decay in the echotrain due to irrecoverable dephasing. The magnetization undergoesvarious magnetic fields as it moves through a gradient picking updifferent phases at each location. The same happens as the tool moves.The magnetization also undergoes different magnetic field strengths viapulling, which mathematically looks exactly the same as a diffusion,

$D = {\frac{\lambda^{2}}{2\; \tau}.}$

Additional de-phasing due to pulling is shown in FIG. 12 with respect topulling the NMR tool 805.

Consider problems with B₁ inhomogeneity. There are inconsistencies inthe B₁ field. The B₁ field naturally falls off in a logging tool as itis facing outward. The higher frequencies tend to be closer to thelogging tool, where the B₁ is stronger. A pulse excites not only thecenter frequency, but a bandwidth around it, BW=1/τ_(pulse). This meansthat the frequencies closer to the tool will tend to tip more than thefurther ones. Because there is a spread of tipping angles, there is somedeparture from the echo prediction equation in itself, but adding motionatop of it, causes even more distortions.

When an inversion is performed on the echoes from a real logging tool,all of these differences from the simplistic view of signal are there.That means, if the data is inverted against a matrix created withequation (1), there can be errors in A0 and in the spectra. Mostly, thiscan cause shifts in to T₁s, and T₂s such that they become shorter andsubsequently for A0 to be over called.

In an embodiment, a solution presented here is to invert against a morerealistic matrix. The NMR tool acquires echoes based on the pulsesequence used, where this data is designated as s(t). To interpret thedata s(t) is inverted into different basis: T₂, T₁, or D. To performthis inversion, the data is fit to known answers.

$\begin{matrix}{{s(t)} = {\sum\limits_{ij}{x_{ij} \cdot {A\left( {T_{1\; i},T_{2\; j}} \right)}}}} & (3)\end{matrix}$

When the tool is stationary, the signal, omitting surface/volumeinteraction, is known to have the form:

$\begin{matrix}{{{jth}\mspace{14mu} {Echo}\text{:}\mspace{14mu} {y^{k}(j)}} = {\sum\limits_{i = 1}^{p}{x_{i} \cdot \left( {1 - ^{- \frac{{WT}_{k}}{T_{1_{i}}}}} \right) \cdot ^{- \frac{j \cdot {TE}}{T_{2_{i}}}} \cdot ^{- \frac{{D{({\gamma \; {GTE}})}}^{2} \cdot j \cdot {TE}}{12}}}}} & (4)\end{matrix}$

which is simplified by relating T_(1i) and T_(2j) by a factor andconsidering only one inner echo time (TE). However, the discussedmethods here can be used for any number of dimensions available in theabove equation (4). Considering the standard T₁ experiment:

$\begin{matrix}{{{nth}\mspace{14mu} {Echo}\text{:}\mspace{14mu} {y^{k}(j)}} = {\sum\limits_{i = 1}^{p}{x_{i} \cdot \left( {1 - ^{- \frac{{WT}_{k}}{T_{1_{i}}}}} \right) \cdot ^{- \frac{j \cdot {TE}}{T_{2_{i}}}} \cdot ^{- \frac{{D{({\gamma \; {GTE}})}}^{2} \cdot j \cdot {TE}}{12}}}}} & (5)\end{matrix}$

Here, WT_(k) is the recovery time, TE is the inter echo spacing, G isthe gradient, T_(1i) and T_(2i) time constants are related, p is thetotal number of T₁ components, and x_(i) is the corresponding amplitude.

The above equation (5) can be written in matrix as below:

$\begin{matrix}{{\begin{matrix}\; \\{{WT}\; 1\mspace{14mu} {data}\mspace{11mu}\{} \\\; \\\; \\{{WT}\; 2\mspace{14mu} {data}\mspace{14mu}\{} \\\; \\\; \\{\mspace{50mu} {{WTm}\mspace{14mu}\{}} \\\;\end{matrix}\;\begin{bmatrix}{y^{1}(1)} \\{y^{1}(2)} \\\ldots \\{y^{1}\left( n_{1} \right)} \\{y^{2}(1)} \\{y^{2}(2)} \\\ldots \\{y^{2}\left( n_{2} \right)} \\\ldots \\{y^{m}(1)} \\{y^{m}(2)} \\\ldots \\{y^{m}\left( n_{m} \right)} \\\ldots\end{bmatrix}} = {\begin{bmatrix}A_{11}^{1} & A_{21}^{1} & A_{p\; 1}^{1} \\A_{12}^{1} & A_{22}^{1} & A_{p\; 2}^{1} \\\ldots & \ldots & \ldots \\A_{1\; n_{1}}^{1} & A_{2\; n_{1\;}}^{1} & A_{{pn}_{1}}^{1} \\A_{11}^{2} & A_{21}^{2} & A_{p\; 1}^{2} \\A_{12}^{2} & A_{22}^{2} & A_{p\; 2}^{2} \\\ldots & \ldots & \ldots \\A_{1\; n_{2}}^{2} & A_{2\; n_{2}}^{2} & A_{{pn}_{2}}^{2} \\\ldots & \ldots & \ldots \\A_{11}^{m} & A_{21}^{m} & A_{p\; 1}^{m} \\A_{12}^{m} & A_{21}^{m} & A_{p\; 1}^{m} \\\ldots & \ldots & \ldots \\A_{1\; n_{m}}^{m} & A_{2\; n_{m}}^{m} & A_{p\; 2n_{m}}^{m}\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\\ldots \\x_{p}\end{bmatrix}}} & (6)\end{matrix}$

where

$A_{ij}^{k} = {\left( {1\; - ^{- \frac{{TW}_{k}}{T_{1_{i}}}}} \right) \cdot ^{- \frac{j \cdot {TE}}{T_{2_{i}}}}}$

is the jth echo associated with the kth recovery time TW_(k), j is theecho number, T_(1i), T_(2i) are corresponding T₁ and T₂ constants, m isthe total number of recovery times, and p the total number of T₁components.

The above equation (6) can be shortened as

$\begin{matrix}{{Y = {AX}},} & \left( {7\text{-}1} \right) \\{where} & \; \\{{Y = \begin{bmatrix}{y^{1}(1)} \\{y^{1}(2)} \\\ldots \\{y^{1}\left( n_{1} \right)} \\{y^{2}(1)} \\{y^{2}(2)} \\\ldots \\{y^{2}\left( n_{2} \right)} \\\ldots \\{y^{m}(1)} \\{y^{m}(2)} \\\ldots \\{y^{m}\left( n_{m} \right)} \\\ldots\end{bmatrix}},{A - \begin{bmatrix}\overset{\rightarrow}{A_{1}^{k}} & \overset{\rightarrow}{A_{2}^{k}} & \ldots & \overset{\rightarrow}{A_{p}^{k}}\end{bmatrix}},{X = \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\\ldots \\x_{p}\end{bmatrix}},{and}} & \; \\{\overset{\rightarrow}{A_{i}^{k}} = {\begin{bmatrix}A_{i\; 1}^{1} \\A_{i\; 2}^{1} \\\ldots \\A_{i\; n_{1}}^{1} \\A_{i\; 1}^{2} \\A_{i\; 2}^{2} \\\ldots \\A_{i\; n_{2}}^{2} \\\ldots \\A_{i\; 1}^{m} \\A_{i\; 1}^{m} \\\ldots \\A_{i\; 2n_{m}}^{m}\end{bmatrix}\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {column}\mspace{14mu} {{vector}.}}} & \;\end{matrix}${right arrow over (A_(j) ^(k))} can be re-written as:

${\overset{\rightarrow}{A_{i}^{k}} = \begin{bmatrix}B_{i}^{1} \\B_{i}^{2} \\\ldots \\B_{i}^{m}\end{bmatrix}},{where}$$\overset{\rightarrow}{B_{i}^{k}} = {\begin{bmatrix}A_{i\; 1}^{k} \\A_{i\; 2}^{k} \\\ldots \\A_{{in}_{4}}^{m}\end{bmatrix}.}$

So, B_(i) ^(k) is echo train with recovery time TW_(k), T₁=T_(1i),T₂=T_(2i), and n_(k) number of echoes, and {right arrow over (A_(i)^(k))} is the combined echo trains with all recovery times TW₁, TW₂, . .. and TW_(mb), but the same T1 and T2. A_(i) ^(k) is called single T1component echo vector.

So the spectrum vector X is given by:

X=A ⁻¹ Y.  (7-2)

The magnetization, M(t), is a function of speed, v, while logging. T₂decay, and T₁ recovery time, B₀ distribution, and initial magnetizationhistory are variables of time, M(t→time). When the tool is moving withspeed v, a single component echo vector {right arrow over (A_(j) ^(k))}varies with speed and is denoted as {right arrow over (A_(j) ^(k)(v))},and corresponding Y becomes Y(v), but X stays unchanged, because X isintrinsic property of formation.

Equation (7-1) can be re-written as

Y(v)=A(v)X,  (8-1)

and the solution X=A(v)⁻¹ Y(v)  (8-2)

If the inverse of stationary matrix A is used in equation (3-2), theresult is

X*=A ⁻¹ Y(v)=A ⁻¹ A(v)X.  (8-3)

Since A(v) is different from the stationary A due to B₀ inhomogeneity,A⁻¹A(v) is not equal to unit matrix I. So, X* is not the same as X. Inanother words, if the stationary matrix A is used with moving data, anerror would be created. This is called motion effect.

Consider corrections to X*. The term X* can be corrected by thefollowing equation:

X=A(v)⁻¹ AX*,  (9)

where A(v)⁻¹A is called the motion correction matrix.

The term A can be calculated directly. As shown above, constructing A(v)matrix under different speed is a key to removing motion effect. Correctinversion can be achieved either by using correct A(v) (motiondependent) in inversion or using A(standard) but with A(v)⁻¹Acorrection.

As shown above, the column vector in A(v) is the combined singlecomponent echo train. If B₀ is known, A(v) matrix can be calculatedeither analytically or through simulation. Because A(v) is a function ofspeed and T₁/T₂ values, the combination of different speed and differentT₁/T₂ value is big, the amount of computation is huge. One way to reducecomputation is to decrease the number of different speeds butinterpolate/extrapolate to correct speed. Another technique is to adjustthe number of T₁/T₂ components (bins).

Consider equation modification. A first approach in creating a differentA(v) can be to modify the predicting equations by multiplying additionalfunctions, which predict these perturbations.

S(t)=E(CPMG)·F(T1 fresh)·D(ROP diffusion)·M(prepolarize)·B(B1)  (10)

The function E(CPMG) represents the losses in the CPMG. For a simplelong, cylindrical, field this can be represented by:

$\begin{matrix}{{E\left( {C\; P\; M\; G} \right)} = {^{- \frac{t}{T\; 2}} \cdot \left( {1 - \frac{R\; O\; P*t}{{Length}\mspace{14mu} {of}\mspace{14mu} {sensitive}\mspace{14mu} {volume}}} \right)}} & (11)\end{matrix}$

The function F(T₁fresh) represents the fresh zone, which appears in a T1experiment during the wait time.

$\begin{matrix}{{F\left( {T\; 1\mspace{14mu} {fresh}} \right)} = {\left( {{\left( {1 - \frac{R\; O\; P*W\; T}{{Length}\mspace{14mu} {of}\mspace{14mu} {sensitive}\mspace{14mu} {volume}}} \right)*\left( {1 - ^{- \frac{WT}{T_{1}}}} \right)} + \left( \frac{R\; O\; P*W\; T}{{Length}\mspace{14mu} {of}\mspace{14mu} {sensitive}\mspace{14mu} {volume}} \right)} \right).}} & (12)\end{matrix}$

The function D(ROP diffusion) is a diffusion factor which takes intoaccount the motion of the tool. The diffusion constant is defined as:

${D = \frac{\lambda^{2}}{2\tau}},$

where λ is the mean distance traveled during time τ. So the diffusionconstant for motion would be:

$\begin{matrix}{{D_{ROP} = {\frac{\left( {{ROP} \cdot {TE}} \right)^{2}}{2 \cdot {TE}} = \frac{{ROP}^{2} \cdot {TE}}{2}}},} & (13) \\{{{D({ROPDlff})} - ^{- \frac{{D{({yOTE})}}^{2}\tau}{12}}},{^{- \frac{\frac{{ROP}^{2} \cdot {TE}}{2}{({yOTE})}^{2}\tau}{12}} - ^{- \frac{{({D + \frac{{ROP}^{2} \cdot {TE}}{2}})}{({yGTE})}^{2}\tau}{12}}}} & (14)\end{matrix}$

The function M(prepolarize) compensates for the over or rarely underpolarization that happens due to the far ends of the magnets. This takessome knowledge of the magnetic field shape. A simplistic one-dimensional(1D) line can be used to represent the field of a tool along its axis asshown in FIG. 13. A more complex two-dimensional (2D) orthree-dimensional (3D) mapping can represent the magnetic field. Thefunction discussed above (equation (2))

M(t _(i+1))=M _(i)+(χB ₀(r,z)−M _(i))·(1−e ^(−Δt/T1))

is then utilized for different T₁s. Using a finite element approach orrepresenting the field as a series of equations, the pre-polarizationcan be found and either held as a table to call or made into a uniqueequation for a particular magnetic field:

Pre-polarization relations can be given by:

$\begin{matrix}{{\frac{M}{t} = {{1/T}\; 1\left( {{\chi \; {B_{0}\left( {t,v} \right)}} - {M(t)}} \right)}}{{\frac{M}{t} + \frac{M}{T\; 1}} = {\chi \; {{B_{0}\left( {t,v} \right)}/T}\; 1}}{{{\frac{M}{t}*^{{t/T}\; 1}} + {M*\frac{\left( ^{{t/T}\; 1} \right)}{t}}} = {\chi \; {{B_{0}\left( {t,v} \right)}/T}\; 1*^{{t/\; T}\; 1}}}{{{\frac{M}{t}*^{{t/T}\; 1}} + {M*\frac{\left( ^{{t/T}\; 1} \right)}{t}}} = {\chi \; {{B_{0}\left( {t,v} \right)}/T}\; 1*^{{{- t}/\; T}\; 1}}}{\frac{\left( {M\; ^{{t/T}\; 1}} \right)}{t} = {\chi \; {{B_{0}\left( {t,v} \right)}/T}\; 1*^{{{- t}/T}\; 1}}}{{{M(t)}^{\frac{1}{T_{1}}}} = {\int{\frac{\chi \; {B_{\varepsilon}\left( {t,v} \right)}}{T\; 1}*^{- \frac{t}{T_{1}}}{t}}}}} & (15)\end{matrix}$

where B₀ is a function of moving speed v and time t, since B₀ isfunction of coordinate R.

Countering the imperfect B₁ is a little more challenging than a simplemultiplier, though one might be created for specific fields, after usingthe single spin simulation, or full volume simulation described below.

Consider single spin simulation. With imperfect B₁, the magnetizationwill generally be under tipped, but on the next pulse it could be overtipped, which may create a systematic noise like raising and lowering ofthe echoes or may cause a simple decrease in echo amplitude as the echotrain proceeds as shown in FIG. 14. Following a single spin through thepulse rotations is the simplest way to understand and predict thebehavior of inhomogeneous B₁.

A vector, M, is designated which represent the magnetization of thesystem. The initial magnitude of this vector can be found using thefunction for P(prepolarize) above. The magnetization will then berotated and allowed to freely precess as it would during the pulsesequence. The “90” and “180” rotations may not be perfect, but may be ofa different value than 90 or 180, which simulates an imperfect B1. Forexample, consider the following

M=0 via chirp

M=M(prepolarize)

M=R N ₉₀ K M

where R_(θ) is a rotation matrix:

$\begin{matrix}{{{R_{x}(\theta)} = \begin{bmatrix}1 & 0 & 0 \\0 & {\cos \; \theta} & {{- \sin}\; \theta} \\0 & {\sin \; \theta} & {\cos \; \theta}\end{bmatrix}}{{R_{y}(\theta)} = \begin{bmatrix}{\cos \; \theta} & 0 & {\sin \; \theta} \\0 & 1 & 0 \\{{- \sin}\; \theta} & 0 & {\cos \; \theta}\end{bmatrix}}{R_{z}(\theta)} = {\begin{bmatrix}{\cos \; \theta} & {{- \sin}\; \theta} & 0 \\{\sin \; \theta} & {\cos \; \theta} & 0 \\0 & 0 & 1\end{bmatrix}.}} & (16)\end{matrix}$

Following the 90 pulse, the magnetization is allowed to freely precess:

M=(R _(FID,θ) M)·[E(CPMG)·F(T1 fresh)·D(ROPdiffusion)·M(polarize)]  (17)

This can be done in one step or more accurately by small Δts which willadd up to the TE/2:

$\theta = {\frac{\Delta \; t}{2}\omega}$ ω = 2π f

Then, the 180 train is repeated for echo train length:

$\begin{matrix}{\mspace{20mu} {{{M = {R_{180} \cdot M}}M} = {\left( {R_{{FID},{{\theta|t} = {TE}}}M} \right) \cdot {\quad{{\left\lbrack {{E({CPMG})} \cdot {F\left( {T\; 1\mspace{14mu} {fresh}} \right)} \cdot {D\left( {{ROP}\mspace{14mu} {diffusion}} \right)} \cdot {M({polarize})}} \right\rbrack \mspace{20mu} \theta} = {{\frac{\Delta \; t}{2}\omega \mspace{20mu} \omega} = {2\pi \; f}}}}}}} & (18)\end{matrix}$

The echo may be recorded for each Δt or solely at the peak of the echoes(increments of TE). It is possible to include a spreading function toadd to this process which might simulate the real system better.

Full simulation of B₀ and B₁ interaction while pulling can be performed.A way of creating “A” more accurately presented herein is to simulatethe echo train completely mathematically through knowledge of the B₀ andB₁. The B₀ and B₁ may be acquired either through simulation or bymeasuring the field from real logging tool with a Gauss meter. The fieldmay be represented as a series of equations along the tool or by cuttingit into some sort of mesh. At each mesh point, the NMR contribution canbe found and it's time dependence calculated as it was for a singlepoint. However, now different B₁ inhomogeneities and NMR frequencies canbe taken into account according to the B₁ and B₀ geometries.

M(t)=s(t)=∫₀ ^(T)local magnetization(t)dt  (19)

FIG. 15 is a pictorial representation of a grid mesh field. Theparameters for this approach may include τ=length of a pulse with abandwidth given by BW=1/τ₁₈₀, center frequency of ω_(center)=−γB₀,minimum frequency of ω_(min)=−2πγB₀−½BW, and maximum frequency ofωmax=−2πγB₀+½BW. At each corner, the B₁ and B₀ would be calculated then,a magnetization calculated, and rotated through time. An echo train canbe followed through this grid.

An example of a simulated echo train is shown in FIG. 16. The inversionof such data is shown in FIG. 17 where the T₁ spectra are shifted toshorter T₁ and where the porosity is calculated too high. Dots in FIG.17 represent where the T₁ should be.

Each point in FIG. 18 is an example of total signal versus T₁ when asingle T1 is inverted using the stationary “A”. Porosity of eachinversion vs T₁ can be conducted using regular SVD inversion. FIG. 19 isan example of a plot of inversion intensity versus T₁ using thesimulated A(v). When the correct “A” is used from a full B₀ and B₁, theinversion has the peaks at the correct T₁s and the porosity is near 100PU where it should be instead of over calling. Dots in FIG. 19 representwhere the T₁ should be with SNR=300. FIG. 20 is a representation showingthe porosity vs T1, using A(v) in inversion. Porosity of each inversionvs T₁ from new inversion no longer overcall the signal.

In various embodiments, a method includes a motion correction thatincorporates the distortions due to T₂ signal loss, T₁ over call, overpolarization, and B₁ inhomogeneities all at once, by either finding acorrective matrix to fix the stationary inversion. In another approach,by knowing the true form the echo train would have with a particular B₁and B₀ and inverting that data, motion correction can be attained.Having a more reliable inversion may give correct porosity and T₂, T₁,and D spectra.

The discussion herein includes a discussion of an embodiment of a noveltechnique for data processing to achieve reliable data while drilling orpulling. This technique accounts for the effects of motion and imperfectRF magnetic field B₁ using simulation of the tool's static magneticfield B₀ and B₁. The technique is demonstrated using T₁ data, but can bealso used with T₂.

NMR logging while drilling (LWD) is an important formation evaluationand geosteering tool that has gained increased use within the industry.The low-magnetic-field gradient NMR while-drilling design has anadvantage of a larger sensitive volume with a small-dimension antennaaperture to potentially provide high-resolution logs. In addition, thewide sensitive volume reduces the detrimental effect of lateralvibration to the NMR measurements. On the contrary, the small apertureantenna is more axial-motion sensitive, especially for T₁ measurementsin a slow-relaxation formation, where a long wait-time (TW) could resultin a nontrivial fresh volume during the time period, rendering incorrectsignal strength and, if uncorrected, can affect the overall porosity andshift the T₁ distribution nonlinearly.

In various embodiments, an LWD NMR tool can be realized by a tool with agradient magnet field varying from of about 1 gauss/cm-6 gauss/cm in thedirection of investigation (DOI) of the sensitive volume. In anotherembodiment, a wireline logging tool could be used, which can be pulledquickly. The wireline logging tool can be realized by a long, highgradient wireline logging tool that can be pulled quickly.

The principle of an NMR experiment is the detection of bulk nuclearmagnetization, which gains a preferred orientation in a magnetic field.This magnetization is manipulated by a RF magnetic field, B1, createdusing a coil.

As previously noted, downhole NMR application can typically use the CPGMecho train pulse sequence as shown in FIG. 6. CPMG consists of anexcitation pulse, commonly used for tipping the magnetization 90°. Asthe free induction decays quickly in a gradient field, a recovery pulse,commonly but not exclusively a 180° pulse, is used to refocus themagnetization, causing a spin echo. The recovery pulse is repeatedseveral times dependent on the experiment and power limitations of thetool. The CPMG has one “excitation” pulse, followed by several recoverypulses. This pulse generally tips the magnetization 90° into thetransverse plane, where it can be detected. As discussed next, thispulse may not be a true 90° pulse, that is, not all of the magnetizationwill tip 100% into the transverse plane at all locations in the excitedzone. Following the excitation pulse is a train of refocusing pulses,typically 180° pulses. The timing between these pulses is known as theinter echo spacing, or echo time (TE). The CPMG is a direct measurementof the transverse magnetization decay (T₂). It is used to indirectlymeasure the longitudinal magnetization buildup (T₁) by varying recoverywait time and to measure diffusion by varying inner echo time. Theactual NMR measurement takes place between pulses and is known as an“echo.”

The theoretical NMR echo signal from a CPMG for a single relaxation timecomponent, is commonly represented by above equation (1):

$\begin{matrix}{{s(t)} = {\left( {1 - ^{- \frac{TW}{T\; 1}}} \right) \cdot ^{- \frac{t}{T\; 2}} \cdot ^{- \frac{D{({\gamma \; {GTE}})}^{2}t}{12}}}} & (1)\end{matrix}$

where, as previously noted, TW (wait time) is the time allowed for themagnetization to polarization, T1 is the longitudinal recovery timeconstant, t is the time of the echo peak, T₂ is the transverse decayconstant, D is the diffusion constant, γ is the gyromagnetic ratio, G isthe gradient, TE is the time between echoes. S/V is the surface tovolume ratio of the pores and ρ is the surface relaxivity. Typicallymeasured data is compared against equation (1) using differenttheoretical T₁ and T₂ to calculate time zero echo amplitude, whichdiscloses total porosity, and to create an inversion spectra with axisof T₁, T₂, and/or D vs amplitude. However, because equation (1) onlycaptures the stationary signal form, it does not take into account themotion while drilling or pulling.

For LWD tools, the ROP is typically between 30 to 120 ft/hr. This motionhas several potential effects on the NMR echo trains to deviate it fromthe previous equation, which causes the inversion to be incorrect.Within an echo train, there are losses attributed to the formationmoving into a new zone while pulsing. In the T₁ experiment, there can beovercall or undercall; because of fresh zones after thesaturation/inversion pulse, there are prepolarization motion dependenteffects caused by magnetic field shape, motion-induced spin-dephase, andadditional tipping errors attributed to imperfect B₁.

It is well-known that the measurement accuracy of NMR tools decreaseswhile being pulled. For tools with very long antenna aperture, thisinaccuracy is small compared to the noise of the downhole toolmeasurement, and is usually either compensated afterward, or simplyignored. Though the inaccuracy of a long antenna tool is widely acceptedthe method presented here could still improve data quality. This isespecially true for when the tool is ran at high speed.

The smaller antenna-aperture associated with a low-gradient LWD NMRtools is more sensitive to the axial motion. Because the sensitivevolume is often not a simple geometry, and the effect of the axialmotion is nonlinear to the distance moved, the correction cannot beeasily applied in the relaxation time domain accurately. For thisreason, a method as taught herein incorporates the speed of axial-motionwith the relaxation decay functions in the inversion coefficient matrix.

At the start of the CPGM is a “90°” excitation pulse. The excited volumeis the only volume that will provide signal from the refocusing pulses.FIG. 21 is a representation of an excited volume, which happens at thetime of the excitation pulse. As the tool moves during the rest of theCPMG the sensitive volume partially moves into a zone with no signalwhich causes the T2 decay to be lower than in the stationary case. So,as the tool moves, illustrated in FIG. 21, the refocusing pulses areapplied on a zone that was not excited by the 90° pulse. This means thatthe T₂ signal is progressively less. During a CPMG echo train, thesignal drops faster than expected because of the motion of the tool.This is because the sensitive volume moves away from the originalexcited zone. This does not cause any error to A0 (maximal signal), butartificially enhances the T₂ decay rate.

The T₁ experiment begins with a saturation pulse. Often, this saturationpulse is a saturation/inversion pulse. This pulse is designed to nullifythe magnetization as the initial state for all experiments withdifferent TWs. This could be thought of as scrambling of themagnetization into all orientations. Instead of a saturation pulse, aninversion pulse, which puts the magnetization into the negativedirection of B₀, could also be used. T₁ experiments measure the rate ofmagnetization buildup in the B₀ direction. This measurement is takenusing indirect measurement by varying the recovery time (i.e., TW (waittime), following a saturation/inversion pulse. Following the recoverytime is a CPMG sequence. The CPMG sequence will suffer the same lossesas previously described. During the wait time, the tool moves, and sodoes the volume of inquiry. When the excitation “90° ” pulse is applied,some of the saturated/inverted zone is no longer in the sensitivevolume, but has been replaced with a 100% polarized volume as shown inFIGS. 22A and 22B, which causes the signal to be higher than expected.This distortion primarily effects the medium wait times.

FIG. 22A is a representation of a T1 sequence showing the movement ofthe sensitive region (excited region) 2244 relative to a nuclearmagnetic resonance tool, where the sensitive volume at the excitationpulse is no longer the sensitive volume that had the saturation pulseapplied. This causes the echo heights to be higher than in thestationary case. Region 2241 represents a saturated/inversioned region.As the tool moves, during the wait time, some fresh zone 2242 moves intothe sensitive region. The zone excited by the excitation pulse (90°pulse) excites both some of the fresh zone 2242 and part of thesaturated/inversioned zone 2241. As the CPMG continues, small portionsof fresh zone 2242 continue to move into the sensitive volume while thesaturated/inversioned zone 2241 moves out. The new zone 2243 during therefocusing pulses does not give signal. FIG. 22B is a cartooned versionof a T₁ experiment considering only the fresh zone 2242 and no signaleffects. Curve 2256 is the actual type signal increase. Curve 2258 isthe theoretical equation (1) growth. The T₂ train decreases faster thanT₂ in equation (1) at all wait times. When the wait time is short, thereis little time for fresh zone accumulation. At middle wait times, 300 to12 ms (depending on T₁), the fresh zone 2242 and saturated/inversionedzone 2241 will separate the most, and the signal will be higher than thetheoretical value in equation (1). At long wait times, 6000 to 18000 ms(depending on T₁), the actual and theoretical value from equation (1)have nearly the same prepolarization.

The majority of logging tools have magnets, which yield stronger B₀field preceding the NMR sensitive volume. The magnetization relaxesaccording to the field strength it detects. For a single T1 component,the magnetization is given by the above equation (2)

M(t _(i+1))=M _(i)+(χB ₀(r,z)−M _(i))·(1−e ^(−Δt/T1))  (2)

It builds up according the field it is currently in and how muchmagnetization currently exists. Prepolarization is affective on amicroscopic level. The micro magnetization, the magnetization at anysmall spot in the formation, will differ because each spot has adifferent history. Essentially, it is a pseudo integration of all themagnetic fields it has experienced until the time of interest. Thismeans it could be higher or lower than if it was polarizing in astationary magnetic field, M=χB₀(1−e^(−Δt/T1)). See FIGS. 23A and 23B.

FIG. 23A is a representation of a curve 2361 showing a magnetic field atconstant distance, r, and a curve 2363 showing magnetization at aconstant speed. The magnetic field is often stronger preceding thesensitive region. Because that tool is being pulled and micromagnetization has memory, this can cause the overall magnetization to belarger when an experiment begins, rather than if the tool were notmoving. Memory in this case is the result of a correct state being thesum of the previous states and interactions.

A diffusion effect created using NMR is caused by the random motion thatoccurs within the formation. It causes an irreversible decay in the echotrain, which is significantly dependent on how fast the magnetization isrefocused. This is attributed to different magnetic fields affecting themagnetization as it moves through a gradient. As the tool moves, themagnetization experiences different magnetic field strengths; which is,mathematically similar to diffusion,

${D = \frac{\lambda^{2}}{2\tau}},$

where λ in diffusion is the root mean square distance travelled duringthe time increment τ. Because the micro magnetization will experiencemore than one B₀ between pulses, the phase of the spins becomes jumbled,just as if it had diffused. FIG. 12 is a representation of additionaldiffusion in nuclear magnetic resonance measurements due to pulling thetool in the borehole. While the tool is moving, a particular part of theformation will undergo the influence of many different B₀s. Thishappening during an experiment causes an unrecoverable phase. This issimilar to the effect of a molecule diffusing through the formation andpicking up additional phase because of the influence of different B₀s.

With respect to problems with B1 Inhomogeneity, there areinconsistencies in the B₁ field. The B₁ field naturally falls off in alogging tool as it faces outward. The higher frequencies (determined byB₀) tend to be closer to the logging tool, where the B₁ is stronger. Apulse excites not only the center frequency, but a bandwidth (BW) aroundit, BW=1/τ_(pulse). This means, within the sensitive volume, there willbe a spread of tipping angles. As a result, there are additionaldepartures from the echo prediction in equation (1).

With respect to stationary inversion, to interpret the measured echoes,s(t), the data is inverted into different basis, T2, T1, or D. Theinversion fits the data to discrete levels of T₁s, T₂s, or Ds, where thecoefficient, X_(ijk), would be the porosity:

s(t)=Σ_(ijk) x _(ijk) ·A(T _(1i) ,T _(2j) ,D _(k))  (20)

Here, A(T_(1i), T_(2j), D) is a theoretically calculated echo train fora specific T₂, T₁, and D. Mapping x_(ijk) against T₁, T₂, or D gives amultidimensional spectrum, while the sum of X_(ijk) is the totalporosity.

The drilling tool, discussed herein, focuses on T1 spectra, which issimplified in practice by relating T_(1i) and T_(2j) by a factor andconsidering only one inner echo time (TE); however, the methods taughtherein can be used for any number of dimensions available in theprevious equation. The stationary known form, inexplicitly including theS/V and diffusion term, of its echo trains can be:

$\begin{matrix}{{{nthEcho}\text{:}\mspace{14mu} {y^{k}(j)}} = {\sum_{i = 1}^{p}{x_{i} \cdot \left( {1 - ^{- \frac{{TW}_{k}}{T_{1_{i}}}}} \right) \cdot ^{- \frac{j \cdot {TE}}{T_{{1_{i}\;/T_{1}}T_{2}{Ratio}}}}}}} & (21)\end{matrix}$

Here, p is the total number of T₁ components, and x_(i) is thecorresponding amplitude. The previous equation can be written as thematrix:

$\begin{matrix}{\mspace{20mu} {{\begin{matrix}\begin{matrix}{{TW}\; 1\mspace{14mu} {data}\mspace{14mu}\{} \\{{TW}\; 1\mspace{14mu} {data}\mspace{14mu}\{}\end{matrix} \\{{TW}\; 1\mspace{14mu} {data}\mspace{14mu}\{}\end{matrix}\left\lceil \begin{matrix}{y^{1}(1)} \\{y^{1}(2)} \\\ldots \\{y^{1}\left( n_{1} \right)} \\{y^{2}(1)} \\{y^{1}(2)} \\\ldots \\{y^{2}\left( n_{2} \right)} \\\ldots \\{y^{m}(1)} \\{y^{m}(2)} \\\ldots \\{y^{m}\left( n_{m} \right)}\end{matrix} \right\rceil} = {\left\lceil \begin{matrix}A_{11}^{1} & A_{21}^{1} & A_{p\; 1}^{1} \\A_{12}^{1} & A_{22}^{1} & A_{p\; 1}^{1} \\\ldots & \ldots & \ldots \\A_{1n_{1}}^{1} & A_{2n_{1}}^{1} & A_{p\; n_{1}}^{1} \\A_{11}^{2} & A_{21}^{2} & A_{p\; 1}^{2} \\A_{12}^{2} & A_{22}^{2} & A_{p\; 2}^{2} \\\ldots & \ldots & \ldots \\A_{1n_{2}}^{2} & A_{2n_{2}}^{2} & A_{{pn}_{2}}^{2} \\\ldots & \ldots & \ldots \\A_{11}^{m} & A_{21}^{m} & A_{p\; 1}^{m} \\A_{12}^{m} & A_{21}^{m} & A_{p\; 1}^{m} \\\ldots & \ldots & \ldots \\\text{?} & \text{?} & \text{?}\end{matrix} \right\rceil \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\\ldots \\x_{p}\end{bmatrix}}}} & (22) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

where

$A_{ij}^{k} = {\left( {1 - ^{- \frac{{TW}_{k}}{T_{1_{i}}}}} \right) \cdot ^{- \frac{j \cdot {TE}}{T_{2i}}}}$

is the jth echo associated with the kth constants, m is the total numberof recovery times, and p the total number of T₁ components. The previousequation can be shortened using:

$\begin{matrix}{\mspace{20mu} {{Y = \left\lceil \begin{matrix}{y^{1}(1)} \\{y^{1}(2)} \\\ldots \\{y^{1}\left( n_{1} \right)} \\{y^{2}(1)} \\{y^{2}(2)} \\\ldots \\{y^{2}\left( n_{2} \right)} \\\ldots \\{y^{m}(1)} \\{y^{m}(2)} \\\ldots \\{y^{m}\left( n_{m} \right)}\end{matrix} \right\rceil},{A = \begin{bmatrix}\overset{\rightarrow}{A_{1}^{k}} & \overset{\rightarrow}{A_{2}^{k}} & \ldots & \overset{\rightarrow}{A_{p}^{k}}\end{bmatrix}},{X = \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\\ldots \\x_{p}\end{bmatrix}},}} & (23) \\{\mspace{20mu} {{{and}\mspace{20mu} \overset{\rightarrow}{A_{i}^{k}}} = {\left\lceil \begin{matrix}A_{i\; 1}^{1} \\A_{i\; 2}^{1} \\\ldots \\A_{i\; n_{1}}^{1} \\A_{i\; 1}^{1} \\A_{i\; 2}^{1} \\\ldots \\A_{i\; n_{2}}^{m} \\\ldots \\A_{i\; 1}^{m} \\A_{i\; 1}^{m} \\\ldots \\\text{?}\end{matrix} \right\rceil \mspace{14mu} {as}\mspace{14mu} {the}\mspace{14mu} {column}\mspace{14mu} {{vector}.}}}} & \; \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

This gives the simple equation, as noted previously in equation (7-1):

Y=AX.  (24)

Rearranging the terms gives, as noted previously in equation (7-2):

X=A ⁻¹ Y,  (25)

the porosity spectra of interest.

With respect to motion inversion, when the tool is moving with speed v,a single component echo vector {right arrow over (A_(j) ^(k))} varieswith speed and is denoted as {right arrow over (A_(j) ^(k)(v))}, andcorresponding Y becomes Y(v), but X stays unchanged, because X is anintrinsic property of the formation. The motion equivalent equation ofY=AX becomes, as noted previously in equation (8-1):

Y(v)=A(v)X  (26)

with the inversion giving the solution, as noted previously in equation(8-2):

X=A(v)⁻¹ Y(v).  (27)

If the inverse of stationary matrix A is used in equation (27), theresulting inversion is incorrect: X*=A⁻¹Y(v)=A⁻¹A(v)X. That is, becauseA(v) is different from the stationary A, attributed to theaforementioned reasons, A⁻¹A(v) is not equal to unit matrix I. So, X* isnot the same as X. In other words, if the stationary matrix A is usedwith moving data, an error is created. The error is caused by the motioneffects described.

The simple description of inversion is only used for illustrating theinversion process. A detailed inversion method used for loggingmeasurements can also include a regularization mechanism to improve thestability of the solution. See, for example, Tannús, A. and Garwood, M.,1997, Adiabatic pulses: NMR in Biomedicine, 10, 423-434; Dunn, K. J.,Bergman, D. J., and LaTorraca, G. A., 2002, Nuclear magnetic resonance:petrophysical and logging applications (handbook of geophysicalexploration: seismic exploration): Pergamon, Danville, Calif., USA; andPrammer, M., 1996, Efficient processing of NMR echo trains: U.S. Pat.No. 5,517,115.

With the complex shapes of the magnetic field, the correct A(v) does nothave a simple analytical form. As shown, constructing A(v) matrix underdifferent speed is the key to removing the motion effect.

In an embodiment, a way of creating A(v) more accurately, as presentedherein, can include procedures to simulate the echo train completelymathematically using the knowledge of the B₀ and B₁. Because thiscalculation is time-consuming, one set of A(v) has been made using thetheoretically perfect B₀ and B₁, instead of by measuring the field fromeach real logging tool with a Gauss meter and creating A(v) for eachtool.

The magnetic and rf fields can be meshed into small voxels, where eachrepresents an amount of micro magnetization. A finer mesh provides moreaccurate echo trains, however becomes more and more computationallychallenging. The tool's response, emf, from any voxel, as in the griddedspace of FIG. 24A, can be calculated using the principle of reciprocity,as outlined in Hoult, D., 2000, The principle of reciprocity in signalstrength calculations—A mathematical guide: Concepts of MagneticResonance, 12, (4), 173-187. Then, the micro magnetizations are followedthrough time using the Bloch equation. In this way, different A(v) canbe found for different ROPs and T1.

A(v,T1,ROP)∝∫_(Volume)micro magentization(t,T1,ROP)dV  (28)

FIG. 24A is a pictorial of a selected sensitive region in a magneticfield. This field is split into many voxels. At each voxel, the B₁ andthe B₀ are used to calculate a signal. That signal is then rotated usingexcitation and refocusing pulses and followed through time to obtain atheoretical echo train. FIG. 24B is a representation of a rotationallysymmetric voxel. This allows a simplification in the calculation to gofrom 3D to 2D using the annulus volume.

In an embodiment, creating the A(v) can be performed using the followingsteps. First, a zero ROP emf is found just as a calibration would beperformed on the tool. This allows to rescale the A(v) into the units ofporosity instead of in voltage units. For this calculation, it issimplest to use the tool as a reference frame, that is, as if the toolwere stationary and the formation continually moving. In this case, themagnetization field is moved in relation to the B₀ and B₁ fields at theROP. The magnetization, M(t), is a function of speed v, T₂ decay, T₁recovery time, B₀, and B₁. A steady state micro magnetization vector iscreated for each volume with the micro magnetization aligned with B₀.The direction of B₀ in each voxel can be designated as: {circumflex over(z)}=[0 0 1].

The sensitive volume is then selected from the B₀ field using the tooloperating frequency. The saturation/inversion pulse is used to null themagnetization in that sensitive volume. Typically, asaturation/inversion pulse can have a bandwidth between ±3 to ±10% ofthe tool's operating frequency. It generally will be larger than theexcitation pulse's bandwidth.

The micro magnetization is then allowed to recover for a total timeequaling the wait time. Recovery occurs in small time increments, Δt,using the following equation:

M _(t) _(i+1) =(M _(t) _(i) +(χB ₀ −M _(t) _(i) ))(1−e ^(−Δt/T) ¹){circumflex over (z)},  (29)

where M is the micro magnetization to be integrated over, and z is themagnetic susceptibility.

The excitation is performed without considering the finite pulse widththat is computed as an instantaneous event. The excitation pulse can bedone with any phase. The refocusing pulse then is best when it isshifted 90° out of phase from the excitation pulse. For example, theexcitation pulse could be along the “x” axis, while the refocusing alongthe “y.” The micro magnetization vector is then rotated using therotation matrix:

$\begin{matrix}{{R_{x}(\theta)} = \begin{bmatrix}1 & 0 & 0 \\0 & {\cos \left( \theta_{tipE} \right)} & {- {\sin \left( \theta_{tipE} \right)}} \\0 & {\sin \left( \theta_{tipE} \right)} & {\cos \left( \theta_{tipE} \right)}\end{bmatrix}} & (30)\end{matrix}$

The tipping angle, θ, for each voxel is determined using the strength ofB₁ perpendicular to B₀, B_(1n), at the voxel's location:

θ_(tipE) =πγB _(1n)τ_(excitation)  (31)

Following the excitation pulse, the micro magnetization is allowed toprecess freely around the static field for ½ TE. From the Blochequation, it is known that magnetization undergoes a precession underthe influence of a magnetic field, which can be simply represented as arotation:

$\begin{matrix}{{R_{z}(\theta)} = \begin{bmatrix}{\cos \left( \theta_{FID} \right)} & {- {\sin \left( \theta_{FID} \right)}} & 0 \\{\sin \left( \theta_{FID} \right)} & {\cos \left( \theta_{FID} \right)} & 0 \\0 & 0 & 1\end{bmatrix}} & (32)\end{matrix}$

During this time, the magnetization is still recovering by means ofequation (31). So, the rotation also performed in small time increments,θ_(FID)=2πγB₀Δt, until a total time of ½TE has passed.

During the FID, it is easy to access that magnetization spreads within acouple hundred microsecond to the point where no signal would bemeasured, making the reason for a recovery pulse obvious.

The first step to recovering the magnetization is to perform a pulse,which will flip the fast and slow spins' phases. Ideally, this would bea 180° pulse; but, as with the excitation pulse, the exact rotation eachvoxel undergoes will be dependent on the B_(1n).

$\begin{matrix}{{R_{y}(\theta)} = \begin{bmatrix}{\cos \left( \theta_{tipR} \right)} & 0 & {\sin \left( \theta_{tipR} \right)} \\0 & 1 & 0 \\{- {\sin \left( \theta_{tipR} \right)}} & 0 & {\cos \left( \theta_{tipR} \right)}\end{bmatrix}} & (33)\end{matrix}$

where θ_(tipR)=πγB_(1n)τ_(recovery). This calculation is repeated forall of the wait times in the T₁ experiment and for T₁ spanning a rangeof time, for example, between 0.01 s and 10 s.

The porosity overcall can be corrected using a more accurate A matrixduring inversion processing. The A matrix, A(v), can be constructed fora number of different speeds with different T1s. Although this techniquerequires long computational time for developing the A(v), it onlyrequires being performed once. For low-gradient logging tools, themethod proves to be very accurate. In tests, data generated had anaverage overcall of 2 PU attributed to the motion effects on a shortaperture. Applications of one or more techniques taught herein may alsoprovide real-time application as data is logged at a well site.

FIG. 25 is a flow diagram of features of an example method to processNMR data taking into account motion of the NMR logging tool. At 2510,echo signals are acquired corresponding to operation of a NMR loggingtool. The NMR can be disposed downhole in a borehole at a drilling siteto evaluate the formation. Theoretical NMR data can be generated withina simulation. At 2520, the echo signals are processed with respect to amatrix that accounts for motion of the nuclear magnetic resonancelogging tool. The matrix can also be based on parameters from theoperation of the nuclear magnetic resonance tool. At 2530, coefficientscorrelated to porosity of the formation are generated.

Various features associated with the method corresponding to FIG. 25 caninclude a number of additional actions or structures. Processing theecho signals can include using a motion dependent matrix constructedunder different speeds of the motion of the nuclear magnetic resonancelogging tool. The motion dependent matrix can be constructed under aselected number of different speeds of the motion of the nuclearmagnetic resonance logging tool augmented with interpolation and/orextrapolation with respect to the different speeds. Processing the echosignals can include using a matrix that accounts for distortions due toT₂ signal loss, T₁ over call, over polarization, and inhomogeneities ofa radio frequency magnetic field B₁ of the NMR logging tool. Processingthe echo signals can include using a matrix that accounts for motion ofthe NMR logging tool stored in a memory system. Processing the echosignals can include processing the echo signals with respect to astationary matrix, the stationary matrix based on parameters from theoperation of the nuclear magnetic resonance tool stationary with respectto the formation; and generating the coefficients correlated to porosityof the formation by applying a motion correction matrix to the echosignals processed with respect to the stationary matrix. The motioncorrection matrix can be realized as the product of an inverse of amotion dependent matrix and the stationary matrix. One of more portionsof these features associated with FIG. 25 may be combined to generateadditional embodiments to process NMR data taking into account motion ofthe NMR logging tool.

FIG. 26 is a flow diagram of features of an example method to processNMR data taking into account motion of the NMR logging tool. At 2610, astationary matrix is generated, where the stationary matrix is based onparameters associated with operation of a nuclear magnetic resonancetool that acquires echo signals to evaluate a formation. At 2620, amotion dependent matrix is formed that accounts for motion of thenuclear magnetic resonance logging tool using the stationary matrix. Themotion dependent matrix can be formed by multiplying each element of thestationary matrix by one or more functions representing effects ofmotion of the nuclear magnetic resonance tool on the respective element.

Various features associated with the method corresponding to FIG. 26 caninclude a number of additional actions or structures. The one or morefunctions can include one or more of a function representing transverserelaxation time (T₂) signal losses, a function representing effects of afresh zone entered by the motion during a wait time, a functionrepresenting a diffusion factor, a function representingprepolarization, or a function representing effects of motion relativeto a radio frequency magnetic field (B₁) of the nuclear magneticresonance logging tool. The multiplication can be realized by a productof the function representing T₂ signal losses, the function representingeffects of a fresh zone entered by the motion during a wait time, thefunction representing a diffusion factor, the function representingprepolarization, and the function representing effects of motionrelative to B₁. The function representing T₂ signal losses can berelated to T₂, rate of penetration of the motion of the nuclear magneticresonance logging tool, and the length of volume sensitive to excitationby the nuclear magnetic resonance logging tool; the functionrepresenting effects of the fresh zone can be related to longitudinalrecovery time (T₁), the rate of penetration of the motion of the nuclearmagnetic resonance logging tool, the length of volume sensitive toexcitation by the nuclear magnetic resonance logging tool, and waittime; the function representing the diffusion factor can be related tothe rate of penetration of the motion of the nuclear magnetic resonancelogging tool; and the function representing prepolarization can berelated to a static magnetic field of the nuclear magnetic resonancelogging tool, the static magnetic field being function of speed andtime. One of more portions of these features associated with FIG. 26 maybe combined to generate additional embodiments to process NMR datataking into account motion of the NMR logging tool.

FIG. 27 is a flow diagram of features of an example method to processNMR data taking into account motion of the NMR logging tool. At 2710, aB₀ and B₁ of a NMR tool are generated. At 2720, the B₀ and B₁ fields aremeshed into a number of voxels. At 2730, response of the NMR tool to theB₀ and B₁ fields are calculated. At 2740, micro magnetizations arefollowed through time. At 2750, a motion dependent matrix is generatedfrom the micro magnetizations, where the motion dependent matrix isoperable to invert signals from nuclear magnetic resonance measurements.

Various features associated with the method corresponding to FIG. 27 caninclude a number of additional actions or structures. Such additionalactions or structures can include, for each waiting time of a pluralityof waiting times and for each longitudinal recovery time (T₁) of aplurality of T₁ times: generating a response of the nuclear magneticresonance tool for zero rate of penetration; generating a steady statemicro magnetization for each voxel aligned with B₀; selecting asensitive volume from the B₀ field using an operating frequency of thenuclear magnetic resonance tool and nulling the magnetization in thesensitive volume; generating an excitation pulse; generating arefocusing excitation pulse; rotating a micro magnetization vector usinga rotation matrix based on a tipping angle for each voxel based of B₁perpendicular to B₀ at the respective voxel location; representingprecession of the micro magnetization following the excitation pulse bya rotation matrix based on magnitude of B₀ and time increments; andrepresenting recover of magnetization by a rotation matrix based on atip angle related to B₁ perpendicular to B₀ at the respective voxellocation and recovery time. One of more portions of these featuresassociated with FIG. 27 may be combined to generate additionalembodiments to process NMR data taking into account motion of the NMRlogging tool.

In various embodiments, features associated with FIGS. 25-27 may becombined to generate a method to process NMR data taking into accountmotion of the NMR logging tool. In addition, a machine-readable storagedevice can have instructions stored thereon, which, when performed by amachine, cause the machine to perform operations, the operationscomprising a method of associated with any of FIGS. 25-27 orcombinations thereof. Further, a machine-readable storage device,herein, is a physical device, which is a non-transitory device, thatstores data represented by physical structure within the device.Examples of machine-readable storage devices include, but are notlimited to, read only memory (ROM), random access memory (RAM), amagnetic disk storage device, an optical storage device, a flash memory,and other electronic, magnetic, and/or optical memory devices.

In various embodiments, a system comprises a nuclear magnetic resonancetool; a control unit coupled to the nuclear magnetic resonance tool tocontrol the nuclear magnetic resonance tool; and a processing unitarranged with the nuclear magnetic resonance tool and a control unit toperform operations according to a method associated with any of FIGS.25-27 or combinations thereof. Various components and/or featuresassociated with the system can include a number of additional structuresor structures arranged to conduct additional actions. The NMR tool caninclude a magnet with an average gradient magnet field varying fromabout 1 gauss/cm to about 60 gauss/cm in a direction of investigation ofa sensitive volume.

FIG. 28 is a block diagram of features of an example embodiment of asystem 2800 operable to process nuclear magnetic resonance data takinginto account motion of a NMR logging tool 2805, as described herein orin a similar manner. The system 2800 can include a NMR tool 2805 havingan arrangement of magnets 2811, antenna(s) 2813, transmitter electronics2812, and receiver electronics 2814. The system 2800 can be configuredto operate in accordance with the teachings herein.

The system 2800 can include a control unit 2825, a memory 2830, anelectronic apparatus 2865, and a communications unit 2835. The memory2830 can be structured to include a database. The control unit 2825, thememory 2830, and the communications unit 2835 can be arranged to operateas a processing unit to control operation of the transmitter electronics2812 and the receiver electronics 2814 and to perform operations on thesignals collected by the receiver electronics 2814 to process nuclearmagnetic resonance data taking into account motion of a NMR logging tool2805. A processing unit 2820, structured to process nuclear magneticresonance data taking into account motion of a NMR logging tool 2805,can be implemented as a single unit or distributed among the componentsof the system 2800 including electronic apparatus 2865. The control unit2825 and the memory 2830 can operate to control activation of thetransmitter electronics 2812 to generate echo train sequences andrecovery pulses. The control unit 2825 and the memory 2830 can operateto control selection of the receiver electronics 2814 in the tool 2805and to manage processing schemes. The control unit 2825, the memory2830, and other components of the system 2800 can be structured, forexample, to operate similar to or identical to the components discussedherein or similar to or identical to any of methods discussed herein.

The system 2800 can also include a bus 2857, where the bus 2857 provideselectrical conductivity among the components of the system 2800. The bus2857 can include an address bus, a data bus, and a control bus, eachindependently configured or in an integrated format. The bus 2857 can berealized using a number of different communication mediums that allowsfor the distribution of components of the system 2800. Use of the bus2857 can be regulated by the control unit 2825. Bus 2857 can include acommunications network.

In various embodiments, the peripheral devices 2845 can includeadditional storage memory and other control devices that may operate inconjunction with the control unit 2825 and the memory 2830. In anembodiment, the control unit 2825 can be realized as a processor or agroup of processors that may operate independently depending on anassigned function. The system 2800 can include display unit(s) 2855,which can be used with instructions stored in the memory 2830 toimplement a user interface to monitor the operation of the tool 2805 orcomponents distributed within the system 2800.

Although specific embodiments have been illustrated and describedherein, it will be appreciated by those of ordinary skill in the artthat any arrangement that is calculated to achieve the same purpose maybe substituted for the specific embodiments shown. Various embodimentsuse permutations and/or combinations of embodiments described herein. Itis to be understood that the above description is intended to beillustrative, and not restrictive, and that the phraseology orterminology employed herein is for the purpose of description.Combinations of the above embodiments and other embodiments will beapparent to those of skill in the art upon studying the abovedescription.

1. A method comprising: acquiring echo signals from operation of anuclear magnetic resonance logging tool to evaluate a formation;processing the echo signals with respect to a matrix that accounts formotion of the nuclear magnetic resonance logging tool, the matrix alsobased on parameters from the operation of the nuclear magnetic resonancetool; and generating coefficients correlated to porosity of theformation.
 2. The method of claim 1, wherein processing the echo signalsincludes using a motion dependent matrix constructed under differentspeeds of the motion of the nuclear magnetic resonance logging tool. 3.The method of claim 2, wherein the motion dependent matrix isconstructed under a selected number of different speeds of the motion ofthe nuclear magnetic resonance logging tool augmented with interpolationand/or extrapolation with respect to the different speeds.
 4. The methodof claim 1, wherein processing the echo signals includes using a matrixthat accounts for distortions due to transverse relaxation time (T2)signal loss, longitudinal recovery time (T₁) over call, overpolarization, and inhomogeneities of a radio frequency magnetic field(B₁) of the nuclear magnetic resonance logging tool.
 5. The method ofclaim 1, wherein processing the echo signals includes using a matrixthat accounts for motion of the nuclear magnetic resonance logging toolstored in a memory system.
 6. The method of claim 1, wherein processingthe echo signals includes: processing the echo signals with respect to astationary matrix, the stationary matrix based on parameters from theoperation of the nuclear magnetic resonance tool stationary with respectto the formation; generating the coefficients correlated to porosity ofthe formation by applying a motion correction matrix to the echo signalsprocessed with respect to the stationary matrix.
 7. The method of claim6, wherein motion correction matrix is the product of an inverse of amotion dependent matrix and the stationary matrix.
 8. A methodcomprising: generating a stationary matrix, the stationary matrix basedon parameters associated with operation of a nuclear magnetic resonancetool that acquires echo signals to evaluate a formation; and forming amotion dependent matrix that accounts for motion of the nuclear magneticresonance logging tool by multiplying each element of the stationarymatrix by one or more functions representing effects of motion of thenuclear magnetic resonance tool on the respective element.
 9. The methodof claim 8, wherein the one or more functions include one or more of afunction representing transverse relaxation time (T2) signal losses, afunction representing effects of a fresh zone entered by the motionduring a wait time, a function representing a diffusion factor, afunction representing prepolarization, or a function representingeffects of motion relative to a radio frequency magnetic field (B₁) ofthe nuclear magnetic resonance logging tool.
 10. The method of claim 9,wherein the multiplication is by a product of the function representingT2 signal losses, the function representing effects of a fresh zoneentered by the motion during a wait time, the function representing adiffusion factor, the function representing prepolarization, and thefunction representing effects of motion relative to B₁.
 11. The methodof claim 9, wherein the function representing T2 signal losses isrelated to T2, rate of penetration of the motion of the nuclear magneticresonance logging tool, and the length of volume sensitive to excitationby the nuclear magnetic resonance logging tool; the functionrepresenting effects of the fresh zone is related to longitudinalrecovery time (T₁), the rate of penetration of the motion of the nuclearmagnetic resonance logging tool, the length of volume sensitive toexcitation by the nuclear magnetic resonance logging tool, and waittime; the function representing the diffusion factor is related to therate of penetration of the motion of the nuclear magnetic resonancelogging tool; and the function representing prepolarization is relatedto a static magnetic field of the nuclear magnetic resonance loggingtool, the static magnetic field being function of speed and time.
 12. Amethod comprising: generating a static magnetic field (B₀) and radiofrequency magnetic field (B₁) of a nuclear magnetic resonance tool;meshing the B₀ and B₁ fields into a number of voxels; calculatingresponse of the nuclear magnetic resonance tool to the B₀ and B₁ fields;following micro magnetizations through time; and generating a motiondependent matrix from the micro magnetizations, the motion dependentmatrix operable to invert signals from nuclear magnetic resonancemeasurements.
 13. The method of claim 12, wherein the method includes,for each waiting time of a plurality of waiting times and for eachlongitudinal recovery time (T₁) of a plurality of T₁ times: generating aresponse of the nuclear magnetic resonance tool for zero rate ofpenetration; generating a steady state micro magnetization for eachvoxel aligned with B₀; selecting a sensitive volume from the B₀ fieldusing an operating frequency of the nuclear magnetic resonance tool andnulling the magnetization in the sensitive volume; generating anexcitation pulse; generating a refocusing excitation pulse; rotating amicro magnetization vector using a rotation matrix based on a tippingangle for each voxel based of B₁ perpendicular to B₀ at the respectivevoxel location; representing precession of the micro magnetizationfollowing the excitation pulse by a rotation matrix based on magnitudeof B₀ and time increments; and representing recover of magnetization bya rotation matrix based on a tip angle related to B₁ perpendicular to B₀at the respective voxel location and recovery time.
 14. Amachine-readable storage device having instructions stored thereon,which, when performed by a machine, cause the machine to performoperations, the operations comprising: acquiring echo signals fromoperation of a nuclear magnetic resonance logging tool to evaluate aformation; processing the echo signals with respect to a matrix thataccounts for motion of the nuclear magnetic resonance logging tool, thematrix also based on parameters from the operation of the nuclearmagnetic resonance tool; and generating coefficients correlated toporosity of the formation.
 15. A system comprising: a nuclear magneticresonance tool; and a control unit coupled to the nuclear magneticresonance tool to control the nuclear magnetic resonance tool; and aprocessing unit arranged with the nuclear magnetic resonance tool and acontrol unit to perform operations to: acquire echo signals fromoperation of the nuclear magnetic resonance logging tool to evaluate aformation; process the echo signals with respect to a matrix thataccounts for motion of the nuclear magnetic resonance logging tool, thematrix also based on parameters from the operation of the nuclearmagnetic resonance tool; and generate coefficients correlated toporosity of the formation.
 16. The system of claim 15, wherein thenuclear magnetic resonance tool has amagnet with an average gradientmagnet field varying from about 1 gauss/cm to about 60 gauss/cm in adirection of investigation of a sensitive volume.
 17. The system ofclaim 15, wherein operations to process the echo signals include use ofa motion dependent matrix constructed under different speeds of themotion of the nuclear magnetic resonance logging tool.
 18. The system ofclaim 15, wherein the motion dependent matrix is constructed under aselected number of different speeds of the motion of the nuclearmagnetic resonance logging tool augmented with interpolation and/orextrapolation with respect to the different speeds.
 19. The system ofclaim 15, wherein operations to process the echo signals include use ofa matrix that accounts for distortions due to transverse relaxation time(T2) signal loss, longitudinal recovery time (T₁) over call, overpolarization, and inhomogeneities of a radio frequency magnetic field(B₁) of the nuclear magnetic resonance logging tool.
 20. The system ofclaim 15, wherein operations to process the echo signals include use ofa matrix that accounts for motion of the nuclear magnetic resonancelogging tool stored in a memory system.
 21. The system of claim 15,wherein operations to process the echo signals include operations to:process the echo signals with respect to a stationary matrix, thestationary matrix based on parameters from the operation of the nuclearmagnetic resonance tool stationary with respect to the formation; andgenerate the coefficients correlated to porosity of the formation byapplication of a motion correction matrix to the echo signals processedwith respect to the stationary matrix.
 22. The system of claim 21,wherein the motion correction matrix is the product of an inverse of amotion dependent matrix and the stationary matrix.
 23. A systemcomprising: a nuclear magnetic resonance tool; and a control unitcoupled to the nuclear magnetic resonance tool to control the nuclearmagnetic resonance tool; and a processing unit arranged with the nuclearmagnetic resonance tool and a control unit to perform operations to:generate a stationary matrix, the stationary matrix based on parametersassociated with operation of the nuclear magnetic resonance tool thatacquires echo signals to evaluate a formation; and form a motiondependent matrix that accounts for motion of the nuclear magneticresonance logging tool by multiplying each element of the stationarymatrix by one or more functions representing effects of motion of thenuclear magnetic resonance tool on the respective element.
 24. Thesystem of claim 23, wherein the one or more functions include one ormore of a function representing transverse relaxation time (T2) signallosses, a function representing effects of a fresh zone entered by themotion during a wait time, a function representing a diffusion factor, afunction representing prepolarization, or a function representingeffects of motion relative to a radio frequency magnetic field (B₁) ofthe nuclear magnetic resonance logging tool.
 25. The system of claim 24,wherein the multiplication is by a product of the function representingT2 signal losses, the function representing effects of a fresh zoneentered by the motion during a wait time, the function representing adiffusion factor, the function representing prepolarization, and thefunction representing effects of motion relative to B₁.
 26. The systemof claim 24, wherein the function representing T2 signal losses isrelated to T2, rate of penetration of the motion of the nuclear magneticresonance logging tool, and the length of volume sensitive to excitationby the nuclear magnetic resonance logging tool; the functionrepresenting effects of the fresh zone is related to longitudinalrecovery time (T₁), the rate of penetration of the motion of the nuclearmagnetic resonance logging tool, the length of volume sensitive toexcitation by the nuclear magnetic resonance logging tool, and waittime; the function representing the diffusion factor is related to therate of penetration of the motion of the nuclear magnetic resonancelogging tool; and the function representing prepolarization is relatedto a static magnetic field of the nuclear magnetic resonance loggingtool, the static magnetic field being function of speed and time.
 27. Asystem comprising: a nuclear magnetic resonance tool; and a control unitcoupled to the nuclear magnetic resonance tool to control the nuclearmagnetic resonance tool; and a processing unit arranged with the nuclearmagnetic resonance tool and a control unit to perform operations to:generate a static magnetic field (B₀) and radio frequency magnetic field(B₁) of a nuclear magnetic resonance tool; mesh the B₀ and B₁ fieldsinto a number of voxels; calculate response of the nuclear magneticresonance tool to the B₀ and B₁ fields; following micro magnetizationsthrough time; and generate a motion dependent matrix from the micromagnetizations, the motion dependent matrix operable to invert signalsfrom nuclear magnetic resonance measurements.
 28. The system of claim27, wherein the operations include, for each waiting time of a pluralityof waiting times and for each longitudinal recovery time (T₁) of aplurality of T₁ times, operations to: generate a response of the nuclearmagnetic resonance tool for zero rate of penetration; generate a steadystate micro magnetization for each voxel aligned with B₀; select asensitive volume from the B₀ field using an operating frequency of thenuclear magnetic resonance tool and nulling the magnetization in thesensitive volume; generate an excitation pulse; generate a refocusingexcitation pulse; rotate a micro magnetization vector using a rotationmatrix based on a tipping angle for each voxel based of B₁ perpendicularto B₀ at the respective voxel location; represent precession of themicro magnetization following the excitation pulse by a rotation matrixbased on magnitude of B₀ and time increments; and represent recover ofmagnetization by a rotation matrix based on a tip angle related to B₁perpendicular to B₀ at the respective voxel location and recovery time.